Problem: Rewrite the following equation in logarithmic form. $ 81=3^{4}$ Rewrite the following equation in exponential form. $ \log_{4}{\left(8\right)}=\dfrac32 $
Solution: The inverse relationship of exponents and logarithms For $m>0$ and $b>0, b\neq 1$, we have the following relationship: $ { b^{{ q}}}}= m $ if and only if $ \log_{ b }{ m}=D q$ Converting the exponential equation So $\, {{3}^{ 4}}}= {81}\,$ implies that $\,\log_{ {3}}({ {81}})=D {4}$. Converting the logarithmic equation Similarly $\, \log_{ {4}}({{8}})=\dfrac32}\,$ implies that $\, {4}^{D { {\frac32}}}={8}$. The logarithmic form of $81=3^{4}$ is: $\log_3{{(81)}}={4}$ The exponential form of $\log_{4}{\left(8\right)}=\dfrac32 $ is: $\,4^{{ {\frac32}}}={8}$